The well-known thermal capillary wave theory, which describes the capillary spectrum of the free surface of a liquid film, does not reveal the transient dynamics of surface waves, e.g. the process through which a smooth surface becomes rough. Here, a Langevin model is proposed that can capture this dynamics, goes beyond the long-wave paradigm which can be inaccurate at the nanoscale, and is validated using molecular dynamics simulations for nanoscale films on both planar and cylindrical substrates. We show that a scaling relation exists for surface roughening of a planar film and the scaling exponents belong to a specific universality class. The capillary spectra of planar films are found to advance towards a static spectrum, with the roughness of the surface $W$ increasing as a power law of time $W\sim t^{1/8}$ before saturation. However, the spectra of an annular film (with outer radius $h_0$) are unbounded for dimensionless wavenumber $qh_0<1$ due to the Rayleigh–Plateau instability.

The steady-state second-harmonic resonance between the fundamental and the second-harmonic modes for waves in a circular basin is investigated by solving the water-wave equations as a nonlinear boundary-value problem. The resulting waves are called (1,2)-waves. The geometry of the basin allows for both travelling waves (TW) and standing waves (SW). A solution procedure based on a homotopy analysis method (HAM) approach is used. In the HAM framework, the mathematical obstacle due to the singularity corresponding to the resonant-wave component can be overcome by adding the resonant term in the initial guess of the velocity potential. Approximate homotopy-series solutions can be obtained for both (1,2)-TW and (1,2)-SW. Two branches of (1,2)-TW and two branches of (1,2)-SW are found. They bifurcate from the trivial solution. For (1,2)-TW, the HAM-based approach is combined with a Galerkin numerical-method-based approach to follow the branches of nonlinear solutions further. The approximate homotopy-series solutions are used as initial guesses for the Galerkin method. As the nonlinearity increases, an increasing number of wave components are involved in the solution.

We investigate the stability of a Newtonian ferrofluid cylinder under a steady magnetic field. Linear stability analysis is performed by imposing a small-amplitude axisymmetric disturbance to the basic state. Admissible magnetic field shapes are found and dispersion relations are obtained for two general shapes: axial and non-axial. For the latter, a wire is needed to avoid a singularity at the cylinder centre. Solutions are plotted for two particular shapes previously discussed in the literature for an inviscid ferrofluid: axial and azimuthal. The applicability of the developed theory to cylinder and jet experiments using a solenoid and a wire carrying a current to create the axial and azimuthal magnetic fields, respectively, is studied. For both cases, results show that the magnetic field has a stabilising effect, as already evidenced. In addition, by solving the equation for the cutoff wavenumber, we show that, unlike the azimuthal case, the axial case cannot be stable for all wavenumbers. Taking into account the ferrofluid viscosity better explains the experimental results on ferrofluid cylinders in an azimuthal magnetic field. By choosing a specific admissible magnetic fields shape, the range of wavenumbers in which the cylinder is unstable can be controlled. A radial magnetic field is one example studied here without providing any details on the means to create the field. This latter case reveals the potential of magnetic fields to control the drop size during the ferrofluid jet breakup. These results may be of interest for applications in the printing and medical fields.

A deep understanding of two-phase displacement in porous media with permeability contrast is essential for the design and optimisation of enhanced oil recovery processes. In this paper, we investigate the forced imbibition behaviour in two dual-permeability geometries that are of equal permeability contrast. First, a mathematical model is developed for the imbibition in a pore doublet, which shows that the imbibition dynamics can be fully described by the viscosity ratio $\lambda$ and capillary number $Ca_m$ which creatively incorporates the influence of channel width and length. Through the finite difference solution of the mathematical model, a $\lambda$–$Ca_m$ phase diagram is established to characterise the imbibition preference in the pore doublet. We then investigate the imbibition process in a dual-permeability pore network using a well-established lattice Boltzmann method, focusing on the competition between the viscous and capillary forces. Like in the pore doublet, the preferential imbibition occurs in high-permeability zone at high $Ca_{m}$ but in low-permeability zone at low $Ca_{m}$. When $Ca_m$ is not sufficiently high, an oblique advancing pattern is observed which is attributed to non-trivial interfacial tension. Using the newly defined capillary number, the critical $Ca_{m}$ curve on which the breakthrough simultaneously occurs in both permeability zones is found to match well with that from the pore doublet and it is the optimal condition for maximising the imbibition efficiency in the entire pore network.